Measuring together with the continuum large Miguel Angel Mota (ITAM) Joint work with David Asper´ o III Arctic Set Theory Meeting
Definition Measuring holds if and only if for every sequence � C = ( C δ : δ ∈ ω 1 ) , if each C δ is a closed subset of δ in the order topology, then there is a club C ⊆ ω 1 such that for every δ ∈ C there is some α < δ such that either • ( C ∩ δ ) \ α ⊆ C δ , or • ( C \ α ) ∩ C δ = ∅ . That is, a tail of ( C ∩ δ ) is either contained in or disjoint from C δ . This principle is of course equivalent to its restriction to club-sequences � C on ω 1 . Measuring is a strong form of failure of Club Guessing at ω 1 . Measuring follows from BPFA and also from MRP .
Definition Measuring holds if and only if for every sequence � C = ( C δ : δ ∈ ω 1 ) , if each C δ is a closed subset of δ in the order topology, then there is a club C ⊆ ω 1 such that for every δ ∈ C there is some α < δ such that either • ( C ∩ δ ) \ α ⊆ C δ , or • ( C \ α ) ∩ C δ = ∅ . That is, a tail of ( C ∩ δ ) is either contained in or disjoint from C δ . This principle is of course equivalent to its restriction to club-sequences � C on ω 1 . Measuring is a strong form of failure of Club Guessing at ω 1 . Measuring follows from BPFA and also from MRP .
Definition Measuring holds if and only if for every sequence � C = ( C δ : δ ∈ ω 1 ) , if each C δ is a closed subset of δ in the order topology, then there is a club C ⊆ ω 1 such that for every δ ∈ C there is some α < δ such that either • ( C ∩ δ ) \ α ⊆ C δ , or • ( C \ α ) ∩ C δ = ∅ . That is, a tail of ( C ∩ δ ) is either contained in or disjoint from C δ . This principle is of course equivalent to its restriction to club-sequences � C on ω 1 . Measuring is a strong form of failure of Club Guessing at ω 1 . Measuring follows from BPFA and also from MRP .
Definition Measuring holds if and only if for every sequence � C = ( C δ : δ ∈ ω 1 ) , if each C δ is a closed subset of δ in the order topology, then there is a club C ⊆ ω 1 such that for every δ ∈ C there is some α < δ such that either • ( C ∩ δ ) \ α ⊆ C δ , or • ( C \ α ) ∩ C δ = ∅ . That is, a tail of ( C ∩ δ ) is either contained in or disjoint from C δ . This principle is of course equivalent to its restriction to club-sequences � C on ω 1 . Measuring is a strong form of failure of Club Guessing at ω 1 . Measuring follows from BPFA and also from MRP .
Theorem (CH) Let κ be a cardinal such that 2 <κ = κ and κ ℵ 1 = κ . There is then a partial order P with the following properties. 1 P is proper. 2 P is ℵ 2 –Knaster. 3 P forces measuring. 4 P forces 2 ℵ 0 = 2 ℵ 1 = κ . 5 P forces b ( ω 1 ) = cf ( κ ) Recall that a poset is ℵ 2 –Knaster iff every collection of ℵ 2 –many conditions contains a subcollection of cardinality ℵ 2 consisting of pairwise compatible cond. Also, b ( ω 1 ) denotes the minimal cardinality of an unbounded subset of ω 1 ω 1 mod. countable.
Theorem (CH) Let κ be a cardinal such that 2 <κ = κ and κ ℵ 1 = κ . There is then a partial order P with the following properties. 1 P is proper. 2 P is ℵ 2 –Knaster. 3 P forces measuring. 4 P forces 2 ℵ 0 = 2 ℵ 1 = κ . 5 P forces b ( ω 1 ) = cf ( κ ) Recall that a poset is ℵ 2 –Knaster iff every collection of ℵ 2 –many conditions contains a subcollection of cardinality ℵ 2 consisting of pairwise compatible cond. Also, b ( ω 1 ) denotes the minimal cardinality of an unbounded subset of ω 1 ω 1 mod. countable.
The theorem will be proved by means of what can be described as a finite support iteration incorporating systems of ctble. struct. with symmetry requirements as side cond. In fact, our forcing P will be P κ , where P κ is the last step of this iteration. The actual construction is a variation of previous works. There are 2 main new ingredients in our present construction. Specifically, at any given stage β < κ of the iteration, (a) the set N q β of models N that are active at that stage, in the sense that β ∈ N and that the marker associated to N at that stage is β , is actually a T –symmetric system (for a suitable predicate T ), and (b) if β = α + 1, we use a separate symmetric system in the working part at α included in the above symmetric system corresponding to the previous stage, i.e., in N q α ; these are the symmetric systems we will denote by O q ,α .
The theorem will be proved by means of what can be described as a finite support iteration incorporating systems of ctble. struct. with symmetry requirements as side cond. In fact, our forcing P will be P κ , where P κ is the last step of this iteration. The actual construction is a variation of previous works. There are 2 main new ingredients in our present construction. Specifically, at any given stage β < κ of the iteration, (a) the set N q β of models N that are active at that stage, in the sense that β ∈ N and that the marker associated to N at that stage is β , is actually a T –symmetric system (for a suitable predicate T ), and (b) if β = α + 1, we use a separate symmetric system in the working part at α included in the above symmetric system corresponding to the previous stage, i.e., in N q α ; these are the symmetric systems we will denote by O q ,α .
This use of local symmetry is crucial in the verification that measuring holds in the final generic extension. Specifically, it is needed in the verification that the generic club C added at a stage α will be such that for every δ ∈ Lim ( ω 1 ) , a tail of C ∩ δ will be contained in C δ in case we could not make the promise of avoiding C δ (where C δ is the δ –indexed member of the club–sequence picked at stage α ).
In a paper from the 80’s, Abraham and Shelah build, given any cardinal λ ≥ ℵ 2 , a forcing notion P which, if CH holds, preserves cardinals and is such that if G is P –generic over V , then in V [ G ] there is a family C of size λ consisting of clubs of ω 1 and with the property that, in any outer model M of V [ G ] with the same ω 1 and ω 2 as V [ G ] , there is no club E of ω 1 in M diagonalising C (where E diagonalising C means that E \ D is bounded in ω 1 for each D ∈ C ). CH necessarily fails in the Abraham–Shelah model V [ G ] since, by a result of Galvin, CH implies that for every family C of size ℵ 2 consisting of clubs of ω 1 there is an uncountable C ′ ⊆ C such that � C ′ is a club.
It is not difficult to see that the generic club added at every stage α < κ of our iteration diagonalises all clubs of ω 1 from V [ G α ] (where G α is the generic filter at that stage). So, it would be impossible to run anything like our iteration over the Abraham–Shelah model without collapsing ω 2 , and therefore we should start from a ground model which is sufficiently different from the Abraham–Shelah model. That is accomplished by imposing that CH must be true in our ground model. Question: Is it consistent to have measuring together with b ( ω 1 ) = ℵ 2 and 2 ℵ 1 > ℵ 2 ?. Important problem: Is measuring compatible with CH?
It is not difficult to see that the generic club added at every stage α < κ of our iteration diagonalises all clubs of ω 1 from V [ G α ] (where G α is the generic filter at that stage). So, it would be impossible to run anything like our iteration over the Abraham–Shelah model without collapsing ω 2 , and therefore we should start from a ground model which is sufficiently different from the Abraham–Shelah model. That is accomplished by imposing that CH must be true in our ground model. Question: Is it consistent to have measuring together with b ( ω 1 ) = ℵ 2 and 2 ℵ 1 > ℵ 2 ?. Important problem: Is measuring compatible with CH?
It is not difficult to see that the generic club added at every stage α < κ of our iteration diagonalises all clubs of ω 1 from V [ G α ] (where G α is the generic filter at that stage). So, it would be impossible to run anything like our iteration over the Abraham–Shelah model without collapsing ω 2 , and therefore we should start from a ground model which is sufficiently different from the Abraham–Shelah model. That is accomplished by imposing that CH must be true in our ground model. Question: Is it consistent to have measuring together with b ( ω 1 ) = ℵ 2 and 2 ℵ 1 > ℵ 2 ?. Important problem: Is measuring compatible with CH?
Notation. if N ∩ ω 1 ∈ ω 1 , then δ N := N ∩ ω 1 . Definition Let T ⊆ H ( θ ) and let N be a finite set of countable subsets of H ( θ ) . We will say that N is a T–symmetric system iff ( A ) For every N ∈ N , ( N , ∈ , T ) ≺ ( H ( θ ) , ∈ , T ) . ( B ) Given distinct N , N ′ in N , if δ N = δ N ′ , then there is a unique isomorphism → ( N ′ , ∈ , T ) Ψ N , N ′ : ( N , ∈ , T ) − Furthermore, Ψ N , N ′ is the identity on N ∩ N ′ . ( C ) N is closed under isomorphisms. That is, for all N , N ′ , M in N , if M ∈ N and δ N = δ N ′ , then Ψ N , N ′ ( M ) ∈ N . ( D ) For all N , M in N , if δ M < δ N , then there is some N ′ ∈ N such that δ N ′ = δ N and M ∈ N ′ .
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